We describe the modulo $2$ de Rham-Witt complex of a field of characteristic $2$, in terms of the powers of the augmentation ideal of the $\mathbb {Z}/2$-geometric fixed points of real topological restriction homology ${\mathrm {TRR}}$. This is analogous to the conjecture of Milnor, proved in [Kat82] for fields of characteristic $2$, which describes the modulo $2$ Milnor K-theory in terms of the powers of the augmentation ideal of the Witt group of symmetric forms. Our proof provides a somewhat explicit description of these objects, as well as a calculation of the homotopy groups of the geometric fixed points of ${\mathrm {TRR}}$ and of real topological cyclic homology, for all fields.

For a not-necessarily commutative ring $R$ we define an abelian group $W(R;M)$ of Witt vectors with coefficients in an $R$-bimodule $M$. These groups generalize the usual big Witt vectors of commutative rings and we prove that they have analogous formal properties and structure. One main result is that $W(R) := W(R;R)$ is Morita invariant in $R$. For an $R$-linear endomorphism $f$ of a finitely generated projective $R$-module we define a characteristic element $\chi _f \in W(R)$. This element is a non-commutative analogue of the classical characteristic polynomial and we show that it has similar properties. The assignment $f \mapsto \chi _f$ induces an isomorphism between a suitable completion of cyclic $K$-theory $K_0^{\mathrm {cyc}}(R)$ and $W(R)$.

We define a theory of Goodwillie calculus for enriched functors from finite pointed simplicial $G$ -sets to symmetric $G$ -spectra, where $G$ is a finite group. We extend a notion of $G$ -linearity suggested by Blumberg to define stably excisive and ${\it\rho}$ -analytic homotopy functors, as well as a $G$ -differential, in this equivariant context. A main result of the paper is that analytic functors with trivial derivatives send highly connected $G$ -maps to $G$ -equivalences. It is analogous to the classical result of Goodwillie that ‘functors with zero derivative are locally constant’. As the main example, we show that Hesselholt and Madsen’s Real algebraic $K$ -theory of a split square zero extension of Wall antistructures defines an analytic functor in the $\mathbb{Z}/2$ -equivariant setting. We further show that the equivariant derivative of this Real $K$ -theory functor is $\mathbb{Z}/2$ -equivalent to Real MacLane homology.